I'm making a temperature and humidity controlled "cave" out of an old fridge/freezer. It seems like both things want about 55F and 70% RH. Is there any reason I couldn't use the same cave for drying both charcuterie and cheese?

you could in theory but bear in mind that you'll probably need differing conditions over the course of drying either your meats or cheese.

I.E. for meats at the beginning stages you want higher RH to prevent case hardening, and lower RH as drying progresses to finish. These numbers are probably still within tolerances for either range of processes.

another thing to consider is that by drying both you will likely have some cross contamination of mold. Cheeses and meats use different strains of mold which will affect the flavors of either.